Probabilistic numerical methods for MFC and MFG based on deep learning
We propose two probabilistic numerical methods for mean field type problems based on deep learning. The first method amounts to solve mean field control problems (i.e., problems of optimal control of McKean-Vlasov dynamics) by learning the optimal control using Monte-Carlo samples and stochastic gradient descent. This can be done in a somewhat brute force fashion thanks to deep learning. The second method deals with forward-backward stochastic differential equation (FBSDE) systems of mean field type. As such, this method can be applied to both mean field control problems and mean field games. We rephrase the problem of finding a solution to a generic mean field FBSDE system as a certain mean field control problem, and we then apply a variant of the first method proposed. Several numerical examples will be provided. This is joint work with René Carmona (Princeton University).